Generalized Raiffa solutions

A. Diskin, M. Koppel, and D. Samet

Abstract

We define a family of solutions for n-person bargaining problems which

generalizes the discrete Raiffa solution and approaches the continuous

Raiffa solution. Each member of this family is a stepwise solution, which

is a pair of functions: a step-function that determines a new disagreement

point for a given bargaining problem, and a solution function that assigns

the solution to the problem. We axiomatically characterize stepwise solu-

tions of the family of generalized Raiffa solutions, using standard axioms

of bargaining theory.

1 Introduction

1.1 Stepwise bargaining solutions

Bargaining is in many cases a process in which the parties achieve interim set-tlements step by step, where each settlement is a starting point for further ne-gotiations. It is possible to describe such a graduated process, in the frameworkof Nash’s bargaining theory (Nash (1950)), as a change of the disagreement,or status quo point, while keeping the set of possible utilities—the bargainingset—fixed. Raiffa (1953) suggested two variants of a solution of this kind, fortwo-player bargaining problems.

In the first solution, the step—the change of the disagreement point—isdiscrete. Given a disagreement point d, the most preferred outcome for a playeris the one that gives her the maximal utility while keeping the other player ather disagreement utility. The new disagreement point is the average of thesetwo, most preferred points. By repeating these steps, the process converges toa Pareto point of the bargaining set.

The second solution suggested by Raiffa is continuous. At each point intime, the process moves continuously in the direction of the average of the twomost preferred points, rather than to this point itself.

Here, we propose a family of discrete solutions that bridges Raiffa’s discretesolution and the continuous one. A generalized Raiffa solution is indexed by p ∈(0, 1]. The new disagreement point for the p-solution is the convex combinationbetween the old disagreement point, and the new disagreement point of theRaiffa solution, where p is the weight of the latter.

For p = 1 we have, of course, the discrete Raiffa solution. Furthermore, aswe will show, when p approaches 0, the solutions approach the continuous Raiffa

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solution. All the solutions we discuss are defined for n players.

1.2 Axiomatization

We provide an axiomatization of the family of generalized Raiffa solutions. Thenovelty of our axiomatization lies in the explicit expression of the graduated as-pect of the bargaining. We propose a stepwise solution for bargaining problemswhich consists of two functions: a single valued step function that determinesthe next disagreement point, and a solution function that assigns the terminalpoint arrived at. The relation between the two function is formulated as anaxiom that says that the solution of a bargaining problem is the solution ofthe problem in which the new disagreement point is determined by the stepfunction.

The axioms used are the most basic and standard axioms of bargainingtheory. Only one axiom is required from the solution function: that it is in-dividually rational. The main axiomatic thrust involves the step function thatdrives the bargaining process. It is required to be strongly individually rational,symmetric, scale covariant, independent of the non-individually rational points,and monotonic in the bargaining set.

These five axioms are almost the same as the axioms defining the Kalaiand Smorodinsky (1975) solution (KS), which also requires Pareto optimality,something that is obviously not required for an interim disagreement point.Symmetry, scale covariance, individual rationality and independence of non-individually rational points are the same in both axiomatizations (though, thelast of these is listed by KS as an “assumption” rather than an axiom). Finally,the monotonicity axiom used here is the same as the simple axiom used in Kalai(1977), and does not require the caveats used in the monotonicity axiom of theKS solution.

1.3 Related work

An axiomatic expression of the process of bargaining was given first in theaxiomatization of the proportional solution in Kalai (1977). The axiom of step-by-step negotiation in Kalai’s paper states that certain types of decompositionsof the bargaining set result in the same solution to the problem. Yet, the axiomdoes not propose a specific process like the ones in Raiffa (1953). Later workemphasizes axioms that involve the change of a disagreement point while keepingthe bargaining set fixed (Thomson (1987), Peters and van Damme (1993), Livne(1989), Anbarci and Sun (2009), Trockel (2009)). In particular, properties ofthe set of disagreement points that do not change the solution were studied insome of this work. In these papers, as well, no assumptions are made regardinga specific process.

Using such properties, Livne (1989) and Peters and van Damme (1993) ax-iomatize the continuous Raiffa solution for two-person bargaining problems.More recently, Anbarci and Sun (2009) propose an axiomatization of the dis-crete Raiffa solution for the two-person case without the explicit use of a step

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function. They, too, axiomatize the solution, but not the process itself.In our axiomatization the process is explicitly assumed. We use a step

function that describes the process by specifying the transition from a givendisagreement point to a new one. This enables us to characterize the generalizedRaiffa solutions for the n-person case, using only “classical” axioms. Unlike theset of all disagreement points that result in the same solution, which is definedin terms of the solution function, the function that determines a single pointthat serves as a new disagreement point cannot be defined in terms of thesolution function. Thus, in our axiomatization this step function constitutes anindependent part of the stepwise solution.

The continuous Raiffa solution is described rigourously and axiomatized byLivne (1989) and Peters and van Damme (1993) for two players only. The paththat leads to the solution is described in the two dimensional space by specifyingthe utility of one player in terms of the utility of the other. We, in contrast,describe a path in the bargaining set parameterized by time. This captures moreclearly the process aspect of the bargaining and is naturally extended to anynumber of players. Although our axioms cannot capture the continuous Raiffasolution, since it does not have a “next disagreement point”, we show that thecontinuous solution is the limit of the discrete generalized Raiffa solutions thatwe characterize. A time parameterized path of partial agreements is describedin O’Neill et al (2004), but bargaining is described there by a continuum ofPareto frontiers rather than one bargaining problem.

2 Preliminaries

Let N be a finite set of players of size n. Utility vectors for N are points inRN . For i ∈ N and x = (xi)i∈N in RN we write x−i for the projection ofx on RN\{i}. We write x = (xi, x−i). For x, y ∈ RN , x ≥ y means that foreach i ∈ N , xi ≥ yi, and x > y means that xi > yi for each i. When weuse elements of RN as linear functionals on RN we denote them in bold face.Thus, for fixed a ∈ RN and α ∈ R, ax = α, where ax =

∑i∈N aixi, describes

a hyperplane in RN . A subset X of RN is comprehensive if for each x ∈ X,{y | y ≤ x} ⊆ X. The interior of a comprehensive set is not empty. The set Xis positively bounded, if there exist a linear functional a ∈ RN and a constantα ∈ R, such that a > 0 and ax ≤ α for each x ∈ X. A point x ∈ X is Pareto

in X, if y ≥ x for y ∈ X implies y = x.A bargaining set for N is a nonempty subset S of RN which is closed, convex,

comprehensive, and positively bounded. In addition we require that all theboundary points of S are Pareto in S.

A bargaining problem (or a problem, for short), for N is a pair (S, d), whereS is a bargaining set and d ∈ S. The point d is called the disagreement point.The individually rational points for a bargaining problem (S, d) are the pointsin Sd = {x | x ∈ S, x ≥ d}. The set of all bargaining problems is denoted by B.

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3 Stepwise-bargaining axioms

A stepwise solution is a pair of functions, (δ, σ), δ : B → RN , σ : B → RN , suchthat for each problem (S, d), δ(S, d) and σ(S, d) are in S. The function δ is thestep function which assigns to each problem a new disagreement point, and σ isthe solution function which assigns to each problem its solution. The followingaxioms should be read with the quantifier “for each (S, d) in B”. The first axiomprovides the justification for calling the pair (δ, σ) a stepwise solution.

Axiom 1 (Step) σ(S, d) = σ(S, δ(S, d)).

There is only one requirement of the solution function: that it selects indi-vidually rational points.

Axiom 2 (σ-Individual rationality) σ(S, d) ∈ Sd.

We now consider several standard axioms, though they involve δ rather thanσ.

Strong individual rationality requires that the new disagreement point isat least as desirable for all players as the old one, and if there is room forimprovement on d, then the new disagreement point should improve upon d.

Axiom 3 (δ-Strong individual rationality) δ(S, d) ∈ Sd, and if d is not

Pareto in S, then δ(S, d) 6= d.

Axiom 4 (δ-Symmetry) If all players are symmetric in (S, d) then they are

also symmetric in δ(S, d).

Axiom 5 (δ-Scale covariance) If a, b ∈ RN , a > 0 and f(x) = (aixi+bi)i∈N ,

then δ(f(S), f(d)) = f(δ(S, d)).

Axiom 6 (δ-Monotonicity) If S ⊆ T then δ(S, d) ≤ δ(T, d)

Axiom 7 (δ-Irrelevance of non-individually rational points) If Sd = Td

then δ(S, d) = δ(T, d)

The family of generalized Raiffa stepwise solutions is introduced in the nextsection. But we already state our main theorem here.

Theorem 1 A stepwise bargaining solution (δ, σ) satisfies axioms 1-7 if and

only if it is a generalized Raiffa solution.

4 Generalized Raiffa solutions

4.1 Defining the solutions

The set of generalized Raiffa solutions is a family of stepwise bargaining solutions(δp, σp) for p ∈ (0, 1]. The solution function σ1 for two players, was introducedin Raiffa (1953).

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For a bargaining set S and x ∈ S the set {yi | (yi, x−i) ∈ S} is notempty since x ∈ S. The boundedness condition on S guarantees that the setis bounded, and by the closedness of S the set is closed. Thus, for a fixedbargaining set S, the following real valued function of x is well defined:

mi(S, x) = max{yi | (yi, x−i) ∈ S}.

The Utopia point for a bargaining problem (S, d) is m(S, d) = (mi(S, d))i∈N .Obviously, m(S, x) ≥ x.

Definition 1 For p ∈ (0, 1] the function δp : B → RN is defined by

δp(S, d) = d + (p/n)(m(S, d) − d).

To see that δp(S, d) ∈ S, consider for each i the point ei = (mi(S, d), d−i)in S. It is easy to see that δ1(S, d) = (1/n)

∑i∈N ei, and as S is convex,

δ1(S, d) ∈ S. For p ∈ (0, 1], δp(S, d) = (1−p)d+p δ1(S, d), and thus δp(S, d) ∈ S.To define σp(S, d), we construct inductively a sequence dk = dk(S, d) of

points in S: d0 = d, and for each k ≥ 0, dk+1 = δp(S, dk). By the definitionof δp, dk+1 ≥ dk, for all k ≥ 0. Since S is positively bounded, the sequence isbounded and therefore there exists a point d∞ = d∞(S, d) such that dk → d∞.

Definition 2 For p ∈ (0, 1] the function σp : B → RN is defined by σp(S, d) =d∞(S, d).

Since S is closed σp(S, d) ∈ S. Thus, the pair (δp, σp) is a stepwise bargainingsolution. We call such a pair a generalized Raiffa solution.

The reason for restricting p to the interval (0, 1] is obvious. For p = 0,δp(S, d) = d and thus no new disagreement point arises. For p > 1, δp(S, d) mayfail to be in S as is the case for (S, 0) where S = {x |

∑i xi ≤ 1}.

4.2 Properties of the solutions

Theorem 2 For each p ∈ (0, 1] and (S, d) ∈ B, σp(S, d) is Pareto in S.

To prove this theorem we first make a simple observation.

Observation 1 For a bargaining set S and x ∈ S, m(S, x) = x iff x is Pareto

in S.

Proof. If x is Pareto in S then for each i there is no point (yi, x−i) in S withyi > xi. Hence, m(S, x) = x. Conversely, if there exists y ∈ S such that y 6= xand y ≥ x, then for some i, yi > xi. By comprehensiveness, (yi, x−i) ∈ S andtherefore mi(S, x) > xi.

We also need the next proposition which will serve us in several theorems.

Proposition 1 For a fixed bargaining set S, m(S, ·) is continuous on S.

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Proof of Theorem 2. For a given (S, d), consider the sequence dk in S thelimit of which is d∞ = δp(S, d). Then, dk+1 = δp(S, dk) = dk +(p/n)(m(S, dk)−dk). Taking the limit and using the continuity of m(S, ·) we have d∞ = d∞ +(p/n)(m(S, d∞) − d∞). Since p > 0 it follows that m(S, d∞) = d∞, and byProposition 1, d∞ is Pareto in S.

Using the continuity of m(S, x) we can show that generalized Raiffa solutionsare continuous in the disagreement point and the parameter p.

Theorem 3 For each bargaining problem S, δp(S, d) and σp(S, d) are continu-

ous in (p, d) ∈ (0, 1] × S.

Except for the first two, all the axioms involve only the step function δ. Thismakes sense, as each stage of the bargaining results solely in reaching a partialagreement over a new disagreement point. Nevertheless, the function σ inheritsall the properties of δ but monotonicity. The proof of the following theorem isstraightforward and is omitted.

Theorem 4 For each p ∈ (0, 1], the solution function σp satisfies axioms 3,4,5,

and 7, mutatis mutandis.

Proposition 1 hinges crucially on the requirement that all boundary pointsare Pareto. The following example demonstrates this point.

Example 1 We construct a set S for n = 3 that satisfies all the requirementsof a bargaining set, except that it has boundary points which are not Pareto.For this set, m(S, ·) is not continuous on S. The construction is based on anexample in Rockafellar (1970) (p. 83).

Consider first the set S0 = {(x1, x2, x3) | x2 < 0, x3 ≤ x21/x2}. It is

straightforward to check that the set is convex. The intersection of S0 withthe plane x2 = a is the hypograph of the parabola x2

1/a that attains its max-imum at 0. As a approaches 0 the parabolas get narrower, converging to theline {(0, 0, b) | b ≤ 0}. The closure of S0 is the union of S0 with this line. Letthe bargaining set S be the comprehensive hull of the closure of S0.

Any point of the form (x1, x2, x21/x2) with x2 < 0 is Pareto in S. In particular

if we choose x1 = 1/n, and x2 = 1/(bn2) for some b < 0, the point yn =(1/n, 1/(bn2), b) is Pareto in S. Thus, m(S, yn) = yn, and limm(S, yn) =(0, 0, b), while m(S, lim yn) = m(S, (0, 0, b)) = (0, 0, 0).

5 The continuous Raiffa solution

Generalized Raiffa solutions are obtained by moving from a given disagreementpoint d to a new one a fixed fraction p/n of the way from d to the Utopia pointm(S, d). When this fraction becomes smaller and smaller the discrete movesconverge to a smooth move. Thus, we can think of a path π(t) in S, whichdescribes a continuous change of disagreement points over time t, and wherethe direction of movement at time t is from π(t) towards m(π(t)). Such a pathdoes indeed exist and it is unique.

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Theorem 5 For each problem (S, d) there exists a unique function π : [0,∞) →S, that solves the differential equation

π′(t) = m(S, π(t)) − π(t),

with the initial condition π(0) = d. The limit limt→∞ π(t), denoted σ0(S, d),exists and it is Pareto in S. The function σ0 is called the continuous Raiffasolution.

Theorem 6 The continuous Raiffa solution is the limit of the the generalized

Raiffa solutions. That is, for each problem (S, d), lim σp(S, d)p→0 = σ0(S, d).

6 Proofs

Proof of Proposition 1. By the convexity of S, mi(S, x) is a finite concavefunction on S. Therefore it is continuous at each point in the interior of S(Theorem 10.1 in Rockafellar (1970)). Assume that x is Pareto in S, thenfor each i, mi(S, x) = xi. Let xk → x. Since mi(S, xk) ≥ xk

i it follows thatlim inf mi(S, xk) ≥ xi. Assume that lim supmi(S, xk) = yi > xi. Then, (yi, x−i)is a partial limit of (mi(S, xk), xk

−i), and as S is closed it is in S. This contradictsthe assumption that x is Pareto in S. We conclude that limmi(S, xk) = xi =mi(S, x). Since each point is either in the interior of S or Pareto in S, the proofis complete.

Lemma 1 If x0 is a boundary point of the bargaining set S and bx ≤ bx0 for

all x ∈ S, then b > 0.

Proof. Let xi < x0i . Then, by comprehensiveness, x = (xi, x

0−i) ∈ S. Since

xi can be negative with a large absolute value, bi ≥ 0. Suppose bi = 0,then bx = bx0. Therefore, x is on the boundary of S. But x is not Pareto,contradicting the requirement that all boundary points of S are Pareto.

Proof of Theorem 1. The function δp(S, d) satisfies the monotonicity axiom(6) since by construction, m(S, d) is monotonic in S. It is easy to see that(δp, σp) satisfies the rest of the axioms.

To show the converse, we first consider bargaining sets defined by hyper-planes. Let S0 = {x |

∑i xi ≤ 1}, and consider the bargaining problem (S0, 0).

By the symmetry axiom (4), there exists p ∈ R such that δi(S0, 0) = p/n for

each i ∈ N . By the strong individual rationality axiom (3), p > 0. Since δ(S0, 0)is feasible, p ≤ 1. Thus, δ(S0, 0) = δp(S0, 0).

Let b > 0 be a functional in RN and β ∈ R. Then S = {x | bx ≤ β} is abargaining set. Consider the problem (S, d) with d that satisfies bd < β. Letai = (β−bd)/bi. Then ai > 0 and therefore aixi +di is a scale transformation.It is easy to see that it transforms (S0, 0) into (S, d). Moreover, the pointδp(S

0, 0) is transformed to d+(p/n)(m(S, d)−d). Thus, by the scale covarianceaxiom (5), δ(S, d) = δp(S, d).

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Next consider any problem (S, d) where d is in the interior of S. This impliesthat mi = mi(S, d) > di for each i ∈ N . Let b be the linear functional b =(1/(mi − di))i∈N and let β = 1 + bd. Consider the problem S = {x | bx ≤ β}for which d is an interior point, and mi(S, d) = mi. Let S− = S ∩ S. It iseasy to check that S− is a bargaining set. Moreover, S−

d = Sd and therefore,

by axiom (7), δ(S−, d) = δ(S, d). We have shown that δ(S, d) = δp(S, d), andthus, by the definition of δp, δp(S−, d) = d + p(m(S, d) − d). We conclude bythe monotonicity axiom (6), that

δ(S, p) ≥ d + p(mi(S, d) − d).

Fix i in N . We construct a problem (S+, d) such that S ⊆ S+ and δi(S+, d) =

δpi (S+, d). Since ei(= ei(S, d)) is a boundary point of S there exists b such that

bx ≤ bei for each x ∈ S. By Lemma 1, b > 0. Consider the bargainingset S+ = {x | bx ≤ bei}. As d is an interior point, bd < bei, and there-fore, as we have shown, δi(S

+, d) = δpi (S+, d) = di + p(mi(S

+

i d) − di). Notethat ei is also a boundary point of S+, and therefore, mi(S

+, d) = mi(S, d).Thus, δi(S

+, d) = di + p(mi(Si, d) − di). But, S ⊆ S+, and therefore by themonotonicity axiom (6),

δi(S, p) ≤ di + p(mi(S, d) − di).

Thus, δi(S, d) = di +p(mi −di) = δpi (S, d). This is true for each i, and therefore

δ(S, d) = δp(S, d).If d is on the boundary of S, then by the definition of a bargaining problem

d is Pareto in S. By the individual rationality axiom (2), δ(S, d) = d and bythe definition of δp, δp(S, d) = d.

We need to show that σ = σp. For (S, d) define a sequence dk by d0 = dand dk+1 = δ(S, dk). By applying the step axiom (1) repeatedly, σ(S, d) =σ(S, dk) for each k. By the individual rationality axiom (2), σ(S, dk) ≥ dk.Hence, σ(S, d) ≥ dk for each k. We have shown that δ(S, dk) = δp(S, dk).Therefore, dk+1 = δp(S, dk), and dk converges to d∞ = σp(S, d). We concludethat σ(S, d) ≥ σp(S, d). But σ(S, d) ∈ S by definition, and σp(S, d) is Paretoby Proposition 1, hence, σ(S, d) = σp(S, d).

Proof of Theorem 3. In view of the continuity of m(S, ·) the functionδp(S, d) = d + (p/n)(m(S, d) − d) is continuous in (p, d). Since S is fixed wemay omit it from some functions. For a given d ∈ S we denote by dk = dk(p, d)the sequence defined by d 0(p, d) = d and dk+1(p, d) = dk + (p/n)(m(dk) − dk).Obviously, d 0 being constant is continuous in (p, d), and by induction for eachk, dk(p, d) is also continuous in (p, d).

Consider neighborhoods in S of the form S+z = {y | y ∈ S, z < y}. For each

p, and d ∈ S+z , δp(S, d) ∈ S+

z , and therefore also σp(S, d) ∈ S+z . Note that for

each y ∈ S+z , y ≤ m(z) by comprehensiveness and as all boundary points are

Pareto, strict inequality must hold. Therefore, S+z = S ∩ {y | z < y < m(z)}.

When x is a boundary point, the open boxes {y | z < y < m(z)} that contain xform a basis for the neighborhoods of x, since when z approaches x from below,m(z) approaches m(x) = x by the continuity of m.

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Fix p ∈ (0, 1] and d ∈ S. We show that σp(S, d) is continuous at (p, d).

Let N be a neighborhood of σp(S, d). Then, N contains a neighborhood of

σp(S, d) of the form S+z . There is a k for which dk(p, d) ∈ S+

z . Choose a

neighborhood N ′ of dk(p, d), contained in S+z . By the continuity of dk, there

exists a neighborhood O of (p, d) such that for each (p, d) ∈ O, dk(p, d) ∈ N ′ ⊆S+

z . Thus, σp(S, dk) ∈ S+z ⊆ N . The proof is complete, as σp(S, d) = σp(S, dk).

Proof of Theorem 5. If d is a boundary point, then π(t) = d for each t and theclaim holds trivially. Let d be a point in the interior of S. The continuity of mon the interior of S guarantees by Peano’s theorem the existence of a solutionπ on some interval [0, ω) (Hartman (1982), Theorem 2.1). For each t in thisinterval, the trajectory from 0 to t is included in a closed bounded subset ofthe interior of S. But in each such set, each function mi satisfies the Lipschitzcondition, by Theorem 10.4 in Rockafellar (1970). Thus, m(S, x) − x satisfiesthe Lipschitz condition on this set, and by the Picard-Lindelof Theorem, thesolution is unique (Hartman (1982)), Theorem 1.1). Finally, let [0, ω) be themaximal interval for the existence of the solution π (where ω is possibly ∞).Then, π(t) converges to the boundary of S when t → ω, by Theorem 3.1 inHartman (1982).

Proof of Theorem 6. Let N be a neighborhood of σ0(S, d). Then, N containsa neighborhood of σ0(S, d) of the form S+

z . There is a t for which π(t) ∈ S+z .

Choose a neighborhood N ′ of π(t), contained in S+z . Note, that the sequence

dk(S, d) defined by d 0 = d and dk+1 = δp(S, dk) = dk + (p/n)(m(S, dk)− dk) isobtained by applying the Euler method to approximate π(t), where p/n is thetime increment. By the continuity of m(S, x)− x the Euler’s method convergesto a solution of the differential equation in the interval [0, t] (See Coddington andLevinson (1955), Theorem 1.2), and since the solution is unique, it converges toπ. In particular, for a small enough p, there exists k such that dk is in N ′ ⊆ S+

z .Therefore, for such p, δp(S, dk) ∈ S+

z . But δp(S, d) = δp(S, dk), which completesthe proof.

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